Hypoelliptic Heat Kernel Over 3-Step Nilpotent Lie Groups

In this paper, we provide explicitly the connection between the hypoelliptic heat kernel for some 3-step sub-Riemannian manifolds and the quartic oscillator. We study the left-invariant sub-Riemannian structure on two nilpotent Lie groups, namely, the (2,3,4) group (called the Engel group) and the (2,3,5) group (called the Cartan group or the generalized Dido problem). Our main technique is noncommutative Fourier analysis, which permits us to transform the hypoelliptic heat equation into a one-dimensional heat equation with a quartic potential.     

Hypoelliptic heat kernel inequalities on the Heisenberg group

We study the existence of “L^p-type” gradient estimates for the heat kernel of the natural hypoelliptic “Laplacian” on the real three-dimensional Heisenberg Lie group. Using Malliavin calculus methods, we verify that these estimates hold in the case p>1. The gradient estimate for p=2 implies a corresponding Poincaré inequality for the heat kernel. The gradient estimate for p=1 is still open; if proved, this estimate would imply a logarithmic Sobolev inequality for the heat kernel.     

The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups

We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector using the Popp's volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.We then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.     

The hypoelliptic Laplacian on a compact Lie group

Let G be a compact Lie group, and let g be its Lie algebra. In this paper, we produce a hypoelliptic Laplacian on G×g, which interpolates between the classical Laplacian of G and the geodesic flow. This deformation is obtained by producing a suitable deformation of the Dirac operator of Kostant. We show that various Poisson formulas for the heat kernel can be proved using this interpolation by methods of local index theory. The paper was motivated by papers by Atiyah and Frenkel, in connection with localization formulas in equivariant cohomology and with Kac's character formulas for affine Lie algebras. In a companion paper, we will use similar methods in the context of Selberg's trace formula.     

Large time behavior for the heat equation on Carnot groups

We first generalize a decomposition of functions on Carnot groups as linear combinations of the Dirac delta and some of its derivatives, where the weights are the moments of the function. We then use the decomposition to describe the large time behavior of solutions of the hypoelliptic heat equation on Carnot groups. The solution is decomposed as a weighted sum of the hypoelliptic fundamental kernel and its derivatives the coefficients being the moments of the initial datum.     

Schatten classes on compact manifolds: Kernel conditions

In this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given.     

On gradient bounds for the heat kernel on the Heisenberg group

It is known that the couple formed by the two-dimensional Brownian motion and its Lévy area leads to the heat kernel on the Heisenberg group, which is one of the simplest sub-Riemannian space. The associated diffusion operator is hypoelliptic but not elliptic, which makes difficult the derivation of functional inequalities for the heat kernel. However, Driver and Melcher and more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel on the Heisenberg group. We provide in this paper simple proofs of these bounds, and explore their consequences in terms of functional inequalities, including Cheeger and Bobkov type isoperimetric inequalities for the heat kernel.     

Positive curvature property for some hypoelliptic heat kernels

In this note, we look at some hypoelliptic operators arising from nilpotent rank 2 Lie algebras. In particular, we concentrate on the diffusion generated by three Brownian motions and their three Lévy areas, which is the simplest extension of the Laplacian on the Heisenberg group H. In order to study contraction properties of the heat kernel, we show that, as in the case of the Heisenberg group, the restriction of the sub-Laplace operator acting on radial functions (which are defined in some precise way in the core of the paper) satisfies a non-negative Ricci curvature condition (more precisely a CD(0,∞) inequality), whereas the operator itself does not satisfy any CD(r,∞) inequality. From this we may deduce some useful, sharp gradient bounds for the associated heat kernel.     

Estimations du Noyau de la Chaleur sur les Espaces Homogènes

We give upper and lower gaussian estimates for the heat kernel for canonical sub-laplacians on homogeneous spaces of Lie groups. We prove that homogeneous spaces of Lie groups of polynomial growth have the doubling property. We deduce Poincaré inequalities on these spaces.

     

Link of groups and homogeneous Hörmander operators

We study a notion of link of Lie groups suggested by the structure of the partial differential operators of Kolmogorov type. As an application of our link procedure we construct explicit examples of stratified Lie groups, with dimension and step arbitrarily large. We also give a set of examples of hypoelliptic second-order operators which are left translation invariant and homogeneous of degree two on the previous groups.

     

Heat kernel analysis on semi-infinite Lie groups

This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the Lp norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds in this setting.     

Trace heat kernel asymptotics in 3D contact sub-Riemannian geometry

In this paper, we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We explicitly compute, in the case of a 3D contact structure, the first two coefficients of the small time asymptotics expansion of the heat kernel on the diagonal, expressing them in terms of the two basic functional invariants χand κ defined on a 3D contact structure.     

A Note on Lower Bounds Estimates for the Neumann Eigenvalues of Manifolds with Positive Ricci Curvature

We study new heat kernel estimates for the Neumann heat kernel on a compact manifold with positive Ricci curvature and convex boundary. As a consequence, we obtain lower bounds for the Neumann eigenvalues which are consistent with Weyl??s asymptotics.     

Global behavior of the heat kernel associated with certain sub-Laplacians on semisimple Lie groups

We prove global sharp estimates for the heat kernel related to certain sub-Laplacians on a real semisimple Lie group, from which we deduce an estimate for the corresponding Green function.     

THE HEAT KERNEL ON THE CAYLEY HEISENBERG GROUP

1IntroductionThe aim of the present paper is to construct an explicit expression of an heat kernel forthe Cayley Heisenberg group of order n.Hulanicki[1]and Gaveau[2]constructed the explicit expression of the heat kernel for theHeisenberg group by using p…     

Approximations for Asian options in local volatility models

We develop approximate formulae expressed in terms of elementary functions for the density, the price and the Greeks of path dependent options of Asian style, in a general local volatility model. An algorithm for computing higher order approximations is provided. The proof is based on a heat kernel expansion method in the framework of hypoelliptic, not uniformly parabolic, partial differential equations.     

Small time heat kernel asymptotics at the cut locus on surfaces of revolution

In this paper we investigate the small time heat kernel asymptotics on the cut locus on a class of surfaces of revolution, which are the simplest two-dimensional Riemannian manifolds different from the sphere with non-trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate point and present the consequences of this result to the small time heat kernel asymptotics at this point. These results give a first example where the minimal degeneration of the asymptotic expansion at the cut locus is attained.     

Heat Kernel Bounds and Homogenization of Elliptic Operators

We obtain upper and lower bounds on the heat kernel of a uniformlyelliptic operator in divergence form with measurable coefficients.From these bounds one can deduce not only the familiar short-timeasymptotics of the heat kernel, but in the case of periodiccoefficients, long-time asymptotics involving homogenizationof the coefficients.     

Hypoelliptic random heat kernels: A case study

We consider the fundamental solution of a simple hypoelliptic stochastic partial differential equation in which the first-order term is modulated by white noise. We derive some short-time asymptotic formulæ. We discover that the form of the dominant short-time asymptotics depends nontrivially upon the interplay between the geometry of the noisy first-order term and the geometry defined by the hypoelliptic operator.